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Pyramidology, a vademecum

Pyramidology refers to various pseudoscientific theories about pyramids, mostly Egyptian, and specifically Khufu’s Great Pyramid of Giza (thereafter the GP). Most of pyramidologists’ claims fall into one or a combination of the following claims: (i) the GP isn’t a tomb but serves other purposes (a power plant, a representation of the Earth, the Solar System, the Milky Way or of the whole Universe, a message intended to us or to somebody else, a giant pyramid-shaped crystal ball…), (ii) it was impossible to build such a monument with known technologies of that time hence, a technologically advanced civilization (Atlanteans, ancient astronauts…) must have did it or, at least, must have helped ancients Egyptians and (iii) there are physical and/or mathematical constants — mostly Pi ($\pi$) and the golden ratio ($\phi$) — intentionally hidden in the dimensions of the GP.

This post focuses on the latest which is arguably the most active and influential branch of pyramidology and serves as a starting point to most pyramidiotic theories. As stated above, most theories focus on $\pi$ and $\phi$: let’s start by this; I’ll try to complete later.

Piramidology and Phiramidology

Piramidology (resp. phiramidology) is the art of looking for Pi ($\pi$) (resp. $\phi$, pronounced “Phi”, the Golden Ratio) in the dimensions of the GP and finding it everywhere. As I will show, piramidology and phiramidology are inextricably linked one another so that one may use these terms interchangeably.

The facts

The GP was designed to be a square pyramid with a total height of 280 royal cubits (c. 146.6 meters) and a square base with edges of 440 royal cubits (c. 230.4 meters). Being a round multiple of 28, the height of the pyramid was probably chosen after the royal cubit which was divided into 7 palms of 4 digits each, yielding a total of 28 digits.

The second key parameter was the 28/22 slope of the GP. It is well documented that ancient Egyptians measured slopes with a seked — that is a number of palms and digits horizontally for one royal cubit vertically. At the time of Khufu, the seked of 5 ½ palms (that is 22 digits) was an obvious choice. The only smooth-sided pyramid that was ever attempted with a shorter seked (resulting in a steeper pyramid) was the so-called Bent Pyramid built by Sneferu, Khufu’s father, with an initial seked of 5 and it was an obvious failure. On the other hand, choosing a longer seked (a flatter pyramid) would have increased construction time and costs dramatically: adding just one digit to the seked of Khufu’s pyramid (a 5 ¾ palm seked) would have increased its volume by 9.3%.

At the time of Khufu, the steeper successful pyramid ever built was the Meidum pyramid. It was originally built as a step pyramid for Khufu’s grandfather Huni and then turned into the first smooth-sided pyramid ever by Sneferu with a seked of 5 ½ palms. So it really was an obvious choice; a choice consistent with virtually everything we know about Egyptian architecture at that time. Applying that seked to the (probably) intended 280 royal cubits height of the GP obviously yields a base with four edges of 440 royal cubits.

It is worth mentioning here that many pyramids where built with that very same seked: the pyramid in Meidum of course but also Menkaure’s pyramid at Giza or Nyuserre’s in Abusir. Starting with the GP it actually became one of the most popular standard for large pyramids and was only surpassed by the 5 ¼ palms seked of Khafra’s pyramid. Shorter seked (larger inclinations) were only used for small pyramids such as satellites.

The bullshit

It happens that $28/22$ (about 1.2727…) is remarkably close to both $4/\pi$ (about 1.2732…) and $\sqrt{\phi}$ (about 1.2720…). It is, of course, a well-known mathematical coincidence but, when manipulated by pyramidologists with basic skills in algebra and presented to plain idiots, it may well serves as a perfect bedrock for a full-scale pyramidiotic theory.

The whole thing may be summarized by the…

First Law of Pyramidology

$$ \frac{4}{\pi} \approx \frac{28}{22} \approx \sqrt{\phi} $$

From this, starting with a guy named John Taylor in 1859, one can easily link the slope of the GP (the $28/22$ ratio) to both $\pi$ and $\phi$ by cherry picking measures and taking advantage of most people's decimal blindness [1].

The most obvious one is, of course, “half the perimeter of the base divided by the height yields $\pi$”. This, of course, uses the $28/22 \approx 4/\pi$ approximate equality. A more sophisticated variation — often used in conjunction with the latest to reinforce the feeling of wonder — is: “the height equals the radius of the circle which circumference equals the perimeter of the base”. Which is the exact same relationship used the other way around. Most (if not all) of piramidology is based that simple relation.

Phiramidology uses the $28/22 \approx \sqrt{\phi}$ approximate equality. Anyone familiarized with the Pythagorean Theorem would inevitably think of the way we measure the hypotenuse of a right triangle. Indeed, “the apothem of each face divided by half an edge of the base yields $\phi$” and alternatively (or cumulatively) “the surface of the four faces divided by the surface of the base yields $\phi$” — which is, of course, the same ratio using areas instead of lengths.

This is pretty much everything one needs to know. The art of piramidology/phiramidology then consist in recycling these approximate equalities, making nice scheme, finding esoteric justifications and mixing them with actual facts: for instance, the floor of the queen’s and the king’s chambers are roughly at 1 and 2 seventh respectively of the total GP’s height (I’ll let you play with that).

Pyrametrology

Pyrametrology is a more subtle and recent form of piramidology/phiramidoly. It is the art of converting royal cubits into meters and then, find $\pi$ and $\phi$ everywhere.

The facts

The main [2] measures of length used by Egyptians at the time of Khufu were the “small” cubit and the royal cubit. The former was divided into 6 palms of 4 digits each while the latter was 1 palm longer. Just like all cubits across history, the origin of the Egyptian cubits is beyond any doubt anthropometric. This is obvious from the names of their subdivisions — digit, palm (4 digits) but also hand (5 digits), fist (6 digits), span (3 palms) — and from the hieroglyph used to write them (there clearly is a human forearm).

While the “small” cubit fits nicely with the forearm, as measured from the elbow to the tip of the middle finger, of a 170 cm tall human (about 45 cm) the royal cubit looks a bit long for a man of that time (about 52.5 cm). It isn’t that surprising: many anthropometric units deemed to be “royal” were greatly exaggerated. The French pied-du-roi (king’s foot), for instance, was well over 32 cm. Whatever the reason, we know the royal cubit was the unit used to build pyramids and that it was already in use around 3000 BC.

The problem is that, in three millennium of history, the actual size of the royal cubit has been varying a lot — so the 52.5 cm figure shouldn’t be understood as a precise number but rather as a raw average. Yet, from the measurements of the GP, most Egyptologists (the real scientists) agree that the royal cubit used in that occasion must have been around 52.35 cm long — the tenth of millimeters, of course, being subject to caution.

The bullshit

And this is where the bullshit starts. It happens that 0.5236, which is a reasonable estimate of the aforementioned royal cubit in meters, is remarkably close to both $\pi/6$ (about 0.523599…) and $\phi^2/5$ (about 0.523607…). It is, again, a well-known mathematical coincidence but you know what pyramidologists do with coincidences — yes, theories.

You saw it coming, here is the…

Second Law of Pyramidology

$$ \frac{1}{6}\pi \approx 0.5236 \approx \frac{1}{5}\phi^2 $$

From this, starting with another guy named Charles Funck-Hellet in 1952, one can easily convert royal cubits into meters to find $\pi$ and $\phi^2$ but, this only works if one follows the two rules of pyrametrology.

Rule #1: any measure in royal cubits that is a multiple of 6 will, once converted into meters, yield a multiple of $\pi$. That is, for any positive integer $f$ one may safely assume that:

$$6f \times 0.5236 \approx f\pi$$

Rule #2: any measure in royal cubits that is a multiple of 5 will, once converted into meters, yield a multiple of $\phi^2$. That is, for any positive integer $f$ the odds are that:

$$5f \times 0.5236 \approx f\phi^2$$

With these two rules and considering the fact that most of the dimensions of the GP in cubits are round numbers, anyone is able to find plenty of $\pi$ and $\phi^2$. For instance: “the height of the pyramid in meters plus half a edge of the base in meters yields a hundred times $\phi^2$.” Check it out yourself. Another one: “the height of the pyramid in meters minus half one edge of the base in meters yields a ten times $\pi$.

And since the “King’s Chamber” is 20 royal cubits long for a width of 10 royal cubits, you won’t be surprised to learn that “the perimeter of the King’s Chamber in meters yields ten times $\pi$” while “the perimeter of the King’s Chamber in meters minus its width in meters yields ten times $\phi^2$.” It works everywhere and, of course, you may use other integers than 10 or 100.

Interim conclusion

So, as you probably have noted, everything we’ve said so far holds together just because of the two Laws of Pyramidology. There’s nothing else. Yet, pyramidologists have discovered a few other amazing coincidences that need to be addressed. I’ll post that here.

To be continued…

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[1] Decimal blindness : Everybody agrees that 3,141.59 and 3,1492.15 are two different figures yet, most people would agree to say that 3.149215 is $\pi$.
[2] There also used — at least — two shorter cubits: the remen (5 palms) and the “sacred cubit” (4 palms).

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